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光场调控技术是当前光学领域的前沿技术及研究热点之一，采用一定的技术手段对激光束的不同维度进行调控可以获得许多具有奇异性质的光场，进一步拓展了其应用[1]。例如，对激光束的频率、时间、复振幅等维度的调控可以获得啁啾脉冲[2]、飞秒激光[3-4]、激光阵列[5-7]等新型光场。对相位维度的调控还可实现多激光束的相干合成以获得高功率密度激光[8-9]。近年来，随着研究的不断深入，对光场的角动量维度进行调控逐渐走入了人们的视野。与宏观物体类似，光子等微观粒子也可携带有角动量。光子的角动量包括自旋角动量（Spin Angular Momentum, SAM）和轨道角动量（Orbital Angular Momentum, OAM），其中光子的SAM仅具有两个本征值（±1），而光子OAM的本征值可为任意整数[10]。事实上，早期人们对光子角动量的研究仅局限于SAM，尽管理论上早已预言了OAM的存在，但对光子OAM的研究一直没有取得实质性的进展。直到1992年，Allen等证明了具有螺旋形相位分布的拉盖尔高斯光束携带有OAM，才开启了光束角动量调控研究的新纪元[11]。起初人们更关注于对光束OAM的调控，获得了具有螺旋形波面且强度分布为中空环形的相位涡旋光束[12-15]，并已在超大容量光通信[16-23]、旋转探测[24-31]、光镊[32-35]、光信息存储[36]、天文技术[37-39]等领域得到应用。后来的研究则拓展至同时调控光束的SAM和OAM，获得具有横截面偏振态各向异性分布的矢量涡旋光束[40-45]，可应用于激光加工[46-47]、高分辨率成像[48]、表面等离子体激发[49]等领域。
对激光SAM和OAM的应用均是建立在知悉所采用的激光束SAM和OAM分布的基础上的，在同一应用场景下，不同的SAM和OAM态分布可带来不同的应用效果，使得测量光束的SAM和OAM分布尤为重要。由于SAM仅具有两个本征值，且其与宏观的偏振态分布有关，因此其测量相对容易。而OAM可具有无穷个本征值，且其主要决定了激光束的波面分布，测量起来较为困难。国内外学者在光束OAM谱的测量方面开展了大量的研究工作，并研发了多种光束OAM谱测量技术。文中主要回顾了近年来光束OAM谱测量技术的发展，同时也将介绍笔者课题组在光束OAM谱测量技术方面所开展的工作。
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携带有OAM是复振幅表达式中包含螺旋相位项exp(ilφ)的光束的固有属性之一，其中l为角量子数，是OAM的本征值，也称为拓扑电荷数或OAM态，φ为角向坐标[11]。OAM光束中所包含的每一个光子均携带有值为l
$\hbar $ （$\hbar $ 为约化普朗克常量）的OAM[11]。一束激光束中可以同时包含有多个不同的OAM成分，这些OAM成分所占的强度比重即OAM强度谱，通常简称为OAM谱。OAM谱决定了光束的光强、相位及波前分布，可以反映激光束的某些特性。先前的研究已经表明，柱坐标系(r,φ,z)下的螺旋谐波$\exp (i l \varphi)$ 是OAM的特征波函数，其中l为任意整数[11]。由于螺旋谐波在角向呈周期性分布，可通过螺旋谐波将光场直接展开。对于任意光场
$u(x, y, {\textit{z}})$ ，其用螺旋谐波$\exp (i l \varphi)$ 展开可得：$$ u(x, y, {\textit{z}})=\frac{1}{\sqrt{2 \pi}} \sum\limits_{l=-\infty}^{+\infty} a_{l}(r, {\textit{z}}) \exp (i l \varphi) $$ (1) 其中，展开系数可以写为：
$$ a_{l}(r, {\textit{z}})=\frac{1}{\sqrt{2 \pi}} \int_{0}^{2 \pi} u(x, y, {\textit{z}}) \exp (-i l \varphi) {\rm{d}} \varphi $$ (2) 由此可得该螺旋谐波上的能量为：
$$ C_{l}=\int_{0}^{\infty}\left|a_{l}(r, {\textit{z}})\right|^{2} r {\rm{d}} r $$ (3) 由于该值不依赖于z坐标，进而可求得该螺旋谐波的相对能量为：
$$ R_{l}=\dfrac{C_{l}}{\displaystyle\sum\limits_{q=-\infty}^{+\infty} C_{q}} $$ (4) 由此可得光束的OAM谱{Rl}。
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现有的光束OAM谱测量方法通常可分为三大类：第一大类即发展最早的衍射测量法，其原理在于设计特殊的衍射光栅，通过分析衍射场的相关性质来反推待测光束的OAM谱[50-70]；第二大类是模式分束法，其中心思想在于通过一定的技术手段将待测光束中不同的OAM成分相互分离，而后再分别测量各个分量的强度得到OAM谱[71-96]；第三大类即除前两类外的其他测量方法，如旋转多普勒法、相干函数法等[97-123]。
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最初的衍射测量方法仅可测量具有单一模式的待测光束的OAM态，当待测光束经特殊设计的衍射光栅时，其远场衍射场会呈现于待测光束OAM态相关的衍射图样。可实现类似功能的衍射光栅包括双缝[51-52]、三角形孔[53-55]、三角形缝[56]、周期渐变光栅及周期渐变衍射器件[57-58]、环形光栅[59]、多点透射光栅[60]等，其对应的远场衍射如图1所示。此外，柱透镜[61-63]和倾斜透镜[64]均可在x和y方向引入不同的光场傅里叶变换，将待测光束转化为类似于厄米高斯光束的光场分布形式，实现和图1(d)~(e)类似的衍射效果，因此也可用来测量具有单一模式的光束的OAM态。
图 1 利用衍射光栅测量单一模式涡旋光束OAM态。(a)双缝衍射[51]；(b)三角形孔衍射[53]；(c)三角形缝衍射[56]；(d)周期渐变衍射器件[58]；(e)环形光栅[59]
Figure 1. OAM state measurement of single mode vortex beams through diffraction gratings. (a) Double-slit diffraction[51]; (b) Triangular aperture diffraction[53]; (c) Annular triangle aperture diffraction[56]; (d) Gradually-changing-period diffraction element[58]; (e) Annular grating[59]
上述技术的共同点在于均将待测光束转化为另一种特殊强度分布的光场，衍射场均存在一主衍射级，使得它们并不适用于测量多模混合光束的OAM态分布及OAM谱。如果设计一种光栅可将基模高斯光束转化为一OAM光束阵列，且该阵列所包含的不同衍射级均对应着不同的OAM态（l∈A，A为各个衍射级OAM态的集合），那么反过来当待测光束入射时，如果其包含的某一OAM态l’满足-l’∈A，则衍射场的阵列中必存在互补OAM态l0，使得l0-l’=0，此时衍射场的光束阵列中必存在OAM态为0的级次，其不再具有中空环形强度分布。此时可通过实心亮斑出现的位置反推出待测光束的OAM态。该方案中，衍射场具有多通道特性，因而可用来测量多模混合光束的OAM态分布。常见的可实现上述功能的光栅包括复合叉形光栅[65-66]、达曼涡旋光栅[67]、整合达曼涡旋光栅[68]等，其对应的OAM态测量范围为−4~4[66]，−12~12[67]及−24~24[68]。
将图像处理技术与衍射光栅相结合即可在OAM态分布测量的基础上实现对待测光束OAM谱的测量。笔者课题组于2016年报道了达曼涡旋光栅衍射场分析的灰阶算法，可用来分析处理待测光束经达曼涡旋光栅后的衍射场直接得出待测光束的OAM谱[69]，如图2所示。灰阶算法的基本原理为：待测光束经达曼涡旋光栅衍射后，其远场衍射中某一衍射级次会出现实心亮斑，而该亮斑完全由待测光束中某一OAM分量转化而来, 因此该实心亮斑的强度可反映出该OAM态在待测光束中的能量比例。该工作采用了参考文献[67]给出的达曼涡旋光栅，其具有25个强度相等的衍射级，且衍射场OAM态分布为−12~12，因此只需逐一扫描各个衍射级中心的实心亮斑强度即可得到待测光束的OAM谱。为了使系统更简单，这里没有使用功率测量设备，而是采用一面阵探测器直接捕获整个衍射场，在不超过面阵探测器阈值的前提下, 各个像素点的灰度值与该像素点响应的光强成正比, 故可用面阵探测器各个像素点输出的灰度值来表征相对光强。根据每一个衍射级中心亮斑区域设置采样范围，并对采样区域被灰度值求和, 则可得到各OAM模式的强度之比。在某些如OAM光通信等应用场景中，待测光束的OAM态分布已知，但还需测量各个OAM分量的强度比重以评估整个系统的性能（如信噪比等）。此时可在参考文献[69]的基础上加以引申，根据待测光束的OAM态分布设计OAM阵列光栅，结合基于灰阶算法的数据实时处理实现OAM模式强度比重的实时监测[70]。
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模式分束法即采用一定技术手段将光束所包含的不同OAM模式在空间范围内按照一定的规律相互分离，且各个模式的排布方式由OAM态决定，通过测量分束得到的各个OAM模式的强度即可得到待测光束的OAM谱。依据分束过程中是否会破坏组成光束的各成分的相位结构，可以将模式分束方法划分为原位型和非原位型。
原位型模式分束的典型之一即基于马赫曾德尔（Mach Zehnder，M-Z）干涉仪的OAM模式分束器，其特点在于在M-Z干涉仪的每条路径上各引入一个道威棱镜，并且两棱镜相对旋转一定的π/2角度，如图3(a)所示。该装置可以作为奇偶OAM态分束器，即奇数阶和偶数阶OAM模式分别从两个端口输出。2002年，Leach等人[71]提出了级联M-Z干涉仪的OAM分束方案，进一步的扩大了分束范围。当级联装置的数量为N时，可分束的OAM模式数为2N，如图3(b)所示，不同阶次的OAM模式将分布从不同的端口输出，值得注意的是每一级所采用的道威棱镜的相对旋转角度α是以0.5倍的方式逐渐减小。2004年，Leach等人在参考文献[71]的基础上，将道威棱镜替换为经特殊设计的光学棱镜，进而将SAM自由度也引入分束中，提出了全角动量（Total Angular Momentum, TAM）的分束技术。同时，该TAM分束方案同样可以通过级联的方式进行扩展[72]。
2011年，Lavery等人设计了一种由两个BS和两个倒相棱镜组合而成的结构，并用实验证明该结构是一种具有鲁棒性的OAM奇偶分束器件[73]。除分束外，将M-Z干涉仪与空间旋转器件耦合还可用来测量OAM谱[74]。2014年，Zhang等人在M-Z干涉分束原理中引入轨道自旋耦合效应，还实现了对分数阶OAM模式的分束[75]。
除M-Z干涉仪法外，还可采用坐标变换方法实现OAM模式分束。理论上该方法只需要两个衍射光学器件，即可将待测光束中的不同OAM成分有效的聚焦在接收平面不同的空间位置以实现OAM模式分束，因此具有较宽的OAM探测动态范围，相比于M-Z干涉仪法有效地简化了系统。需注意的是，经过坐标变化法分束后光场各OAM模式成分的螺旋相位被破坏，即分束后光场中不再具有中空环形的强度分布，因此是一种非原位OAM模式分束技术。2010年，Berkhout等人首次报道了采用坐标变化法分离OAM模式的工作[76]。如图4所示，设计了坐标变换光栅与补偿光栅，其中前者用以将极坐标转换为对数坐标，而后者则在修正坐标转换的过程中引入相位扭曲，进而将入射光束的螺旋相位转化为倾斜平面相位，而后用一透镜聚焦，此时位于透镜像方焦平面内的光场由不同OAM模式形成的条状光斑构成，这些条状光斑的位置由其对应的OAM态决定，其中中心位置为0阶，正负OAM模式分别位于0阶光斑两边，越远离中心衍射级，OAM阶次的绝对值越大。需要指出的是该方法虽能实现任阶次的OAM模式的分束，但是由于分束后相邻OAM成分之间存在重叠，使得无法将模式间隔较小的OAM成分完全分开，这意味着采用该技术测量光束的OAM谱将出现较大的误差。因此为了准确测得OAM谱，还需在此基础上继续优化。
图 4 基于坐标变换的OAM模式分束技术。(a)模式分束光栅；(b)补偿光栅；(c)不同模式分布的待测光束入射时，数值仿真得到的入射位置处、过补偿光栅处、系统接收平面处的强度分布，以及对应的实验测得的接收平面处的强度分布[76]
Figure 4. OAM mode sorter from Cartesian to log-polar coordinate transformation. (a) Coordinate transforming grating; (b) Phase-correcting grating; (c) Modeled and observed intensity profiles at before the transforming optical grating, just after the phase-correcting grating, and the modeled and observed images in the detector plane[76]
2013年，Mirhosseini等提出了一种坐标变换优化方法，将补偿光栅后的光场波前复制多份，扩大了线段形光斑的横向长度，使得经透镜聚焦后由各个OAM模式转化而来的条形光斑更细，进而消除了相邻OAM模式间的光场重叠[77]，如图5(a)所示。需注意的是，虽然直接扩大入射光束的尺寸可扩大补偿光栅后的线段形光斑以使分束后单个模式的条纹更细，但由于相邻模式之间的间隔也相应减小，相邻模式间的重叠依然存在[77-78]。2017年，Li等人在坐标变换的基础上，对待测光场的径向方向引入了2πm(lnr/lnR)的额外的相位，其中m为引入相位的空间频率，r为径向坐标，R为入射光的最大半径，使得分束后接收平面的光场中不同的OAM成分同时存在横向和纵向的平移，其中横向位置仍由OAM态l决定，纵向位置则由参数m决定[79]。由于相邻OAM模式之间横纵方向均存在间隔，使得相邻模式间隔变大进而消除了相邻模式间的重叠，如图5(b)所示。2018年Wen等提出了“螺旋解环”法，如图5(c)所示，该方法的特点在于：“螺旋解环”的螺旋线（图5(c)中的红线）可在光束宽度不受限的理想情况下沿着角向无限延伸，使得补偿光栅后线段形光场具有更大的相位梯度，因此在聚焦后相邻模式间具有更大的空间距离，进而消除了相邻模式间的重叠[80-81]。参考文献[77-80]提出的技术方案可以在一定程度上改善坐标变换法的分束效果，但在实际可分束OAM模式范围仍然受到衍射器件分辨率等多方面因素限制。2013年，Lavery等人指出参考文献[76]报道的坐标变换OAM分束方法的可分束OAM模式阶次l须满足|l|<<2πr2/(Lλ)，其中r为入射光半径，L为模式分束光栅与补偿光栅间的光学距离，因此他们改变了光栅尺寸，并减小了光栅之间的距离L，此时r2/(Lλ)≈760，实现了OAM态范围为−28~28的OAM模式分束[82]。
在坐标变换OAM分束的基础上，分别测量各个模式的强度即可得待测光束的OAM谱[83]。此外还可采用微纳加工等方式基于亚克力等材料加工上述模式分束光栅及补偿光栅以制得模式分束器[84-87]，以适应光子芯片、硅基光子学等应用场景。2018年，Ruffato等人报道了紧凑型OAM模式分束器，通过一个衍射光学器件实现了多个衍射器件的功能，即外侧用于对入射光束坐标变换和波前相位调制，内侧用于相位补偿，同时有效减弱了相邻模式间的重叠，模式间串扰减小至−10 dB[88]。此外，亦可采用超材料、液晶等光学各向异性材料以几何相位的方式实现模式分束光栅和补偿光栅的功能，进而制得光束TAM分束器[89-96]。
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除了衍射测量法和模式分束法外，国内外还报道了多种其他OAM谱测量方法，如角相干函数法[97-101]、旋转多普勒频移法[102-105]、时间映射法[106-108]、单像素成像法[109-115]、相关滤波法[116-121]、高阶强度矩法[122-123]等。
角相干函数法即通过测量待测光场的角相干函数并结合反傅里叶变换重建待测光场的OAM谱，其中角相干函数可由M-Z干涉仪测量干涉对比度间接测得[97-98]。在测量过程中需在M-Z干涉仪的某一臂引入变化旋转角度以测得干涉对比度相对于旋转角度的函数关系，使得在整个测量过程中必须保证严格的光路对准。该OAM谱测量技术中，还可采用角向双狭缝代替M-Z干涉仪，可在一定程度上降低对系统的光路对准要求，但仍然需要进行多次测量，并且由于角向狭缝只利用了很小一部分待测光场，对较低强度入射光场的测量能力有限[99-101]。旋转多普勒频移法即基于携带有OAM的光束的旋转多普勒效应，但待测光束与探测器存在相对同轴旋转运动时，待测光束的频谱会出现频移成分，且该成分的频移量与待测光束的OAM态相关[102-103]。因此可通过分析回波信号的旋转多普勒频移来反推待测光束的OAM谱。2003年，Vasnetsov等人提出了利用旋转多普勒频移测量光束的OAM谱的技术方案[104]。2017年，Zhou等人则基于旋转多普勒频移报道了OAM复振幅谱测量技术[105]，实现了OAM功率谱和相位谱的同时测量，如图6(a)所示。2011年，Bierdz等人基于OAM与时间的映射关系，利用反事实测量和量子芝诺效应，在忽略光学损耗或失调的情况下以100%的效率将任意输入光脉冲的不同OAM分量映射到输出端的不同时间单元，进而设计了一紧凑的OAM谱测量系统[106]。2012年，Karimi等设计了如图6(b)所示的系统，他们将q波片置入环路中，而后采用模式滤波器测量输出基模成分，以此建立待测光束各个OAM模式与输出基模脉冲间隔的函数关系，通过脉冲的延迟时间反推待测光束的OAM成分[107]。Bierdz等人在此基础上提出结构更加简单的基于OAM时间映射的OAM谱测量技术[108]。单像素成像技术已经被证明可重建光场的复振幅分布[109-111]，在重建复振幅的基础上即可根据公式(1)~(4)测量光束的OAM谱。2018年，Ota等展示了基于单像素相机的复振幅成像方法，并对相位重建结果做了定量分析，准确度达到λ/63[112]。2019年，Liu等设计了一种类似于棋盘结构的掩摸，从而避免了引入额外的参考光，可适用于任意强度分布的复振幅光场的重建[113-114]。2020年，同一课题组[115]又提出利用单像素成像技术重建OAM光束波前的工作，并在此基础上测得了待测光束径向量子数功率谱、OAM功率谱及相位谱，如图6(c)所示。相关滤波法即基于模式解调理论合理地选择基底，使得待测光场解调至该基底的不同模式成分互不影响，进而测得OAM谱[116-121]。2018年，Volyar等人采用如图6(d)所示的系统，通过测量待测光场及其经柱透镜衍射后的光场分布得到高阶强度矩，此时可以建立N个不同OAM模式的复系数与高阶强度矩相关的方程，联立这N个方程即可求解各OAM模式成分的复系数，进而测得OAM谱[122]。采用该技术还可测量光束的径向量子数功率谱[123]。
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笔者课题组于2020年报道了光束OAM谱的通用测量技术，其可适用于对任意光场分布光束的OAM谱测量[124]。如图7(a)所示，该技术引入一参考高斯光束ER与待测光束E同轴干涉，并采用一面阵探测器分别测量待测光束、参考高斯光束、干涉光束的强度分布I=|E|2，IR=|ER|2及Icos=|E + ER|2，而后再为参考光束引入π/2相位延迟，测得干涉光束的光场分布Isin=|E+ERexp(iπ/2) |2，此时通过I，IR，Icos和Isin可反演待测光束的复振幅分布。则根据公式(1)~(4)即可算出待测光束的OAM谱。该技术的本质基于光场的螺旋谐波展开，因此其对任意光束均有效，是一种通用的OAM谱测量技术。
该工作针对不同光场分布的待测光束做了大量的测试，其中图7(b)及7(c)给出了双模混合多环涡旋光束的测试结果。该测试采用级联双液晶空间光调制器的方法生成了双模混合多环涡旋光束，并可通过调节两个调制器间半波片的快轴角度θ改变两个OAM模式间的强度比例。图7(b)为实验测得的待测光束的光场分布及其对应的数值仿真结果，图7(c)为OAM谱测量结果，不难看出，实验结果与仿真结果基本吻合，但仍存在一定差异，其原因在于该技术中复振幅的反演是通过图像处理面阵探测器测得的强度分布完成的，使得测量精度与所使用的面阵探测器的分辨率相关，采用高分辨率面阵探测器可有效减小OAM谱测量误差。
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文中主要回顾了近年来光束OAM谱测量技术的国内外进展，重点介绍了衍射光栅法、模式分束法等OAM谱测量方法，此外还介绍了光束OAM谱的通用测量技术。光束OAM谱测测量是OAM应用的重要基础之一，现阶段的OAM谱测量技术仍存在测量范围较小、系统体积较大等问题，因此研发小型化、紧凑型、高OAM态测量范围的OAM谱测量系统是未来OAM探测技术的发展方向之一。
Advances on the measurement of orbital angular momentum spectra for laser beams (Invited)
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摘要: 自Allen等证明具有螺旋相位波面的激光束携带有轨道角动量以来，对光束轨道角动量调控技术的研究取得了跨越式的发展，获得了包括相位涡旋光束、矢量涡旋光束、激光束阵列等多种新型结构光场，在超大容量光通信、遥感探测、激光加工、高分辨率成像等领域展现出广阔的应用前景。准确测量光束的轨道角动量是其应用的重要基础，早期人们更多地关注对待测光束所包含的轨道角动量成分分布的测量，后来逐步拓展至对各个轨道角动量成分的强度比重即轨道角动量谱的测量。文中系统地回顾并总结近年来光束轨道角动量谱测量技术的发展，主要介绍了包括基于衍射、模式分束等方法的新型光束轨道角动量谱测量技术。Abstract: Since Allen et al. have shown that laser beams with helical wavefront carry orbital angular momentums (OAMs), great advances have been achieved for manipulating beams’ OAMs, and contribute to lots of novel structured beams as optical phase and polarization vortices, laser beam lattices. Such structured fields can find applications in lots of domains including large-capacity data-transmission, remote detection, laser manufacture, high-resolution imaging. One of the important bases of above scenarios is diagnose the OAM spectrum. In the early stage, researchers concentrate more on the measurement of OAM distributions, and afterwards expanded gradually to the intensity proportion measurement of each OAM component, namely the orbital angular momentum spectrum. In this paper, the recent advances of OAM spectrum measurement for laser beams were systematically reviewed and summarized, covering approaches of OAM spectrum measurement based on diffraction, mode sorting and other novel methods.
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图 1 利用衍射光栅测量单一模式涡旋光束OAM态。(a)双缝衍射[51]；(b)三角形孔衍射[53]；(c)三角形缝衍射[56]；(d)周期渐变衍射器件[58]；(e)环形光栅[59]
Figure 1. OAM state measurement of single mode vortex beams through diffraction gratings. (a) Double-slit diffraction[51]; (b) Triangular aperture diffraction[53]; (c) Annular triangle aperture diffraction[56]; (d) Gradually-changing-period diffraction element[58]; (e) Annular grating[59]
图 4 基于坐标变换的OAM模式分束技术。(a)模式分束光栅；(b)补偿光栅；(c)不同模式分布的待测光束入射时，数值仿真得到的入射位置处、过补偿光栅处、系统接收平面处的强度分布，以及对应的实验测得的接收平面处的强度分布[76]
Figure 4. OAM mode sorter from Cartesian to log-polar coordinate transformation. (a) Coordinate transforming grating; (b) Phase-correcting grating; (c) Modeled and observed intensity profiles at before the transforming optical grating, just after the phase-correcting grating, and the modeled and observed images in the detector plane[76]
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